Use of quasi-static nanoindentation data to obtain stress ... · Use of quasi-static nanoindentation data to obtain stress–strain characteristics for metallic materials J. Dean,

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Acta Materialia 58 (2010) 3613–3623

Use of quasi-static nanoindentation data to obtainstress–strain characteristics for metallic materials

J. Dean, J.M. Wheeler, T.W. Clyne *

Department of Materials Science & Metallurgy, Cambridge University, Pembroke Street, Cambridge CB2 3QZ, UK

Received 30 December 2009; received in revised form 17 February 2010; accepted 19 February 2010Available online 19 March 2010

Abstract

This paper concerns optimization of procedures and algorithms for extraction of stress–strain relationships from quasi-staticnanoindentation experiments, using finite element method modelling. Several issues are highlighted, including the usefulness ofincorporating residual indent shape in the comparisons, as well as load–displacement–time data, and the significance of creepand interfacial friction. The study is focused on extruded copper bar, using a spherical indenter and assuming transverse isotropythroughout. It is shown that, using the methodology presented here, experimental nanoindentation data could be used to estimatethe yield stress and work-hardening rate, with good accuracy, i.e. the yield stress could have been obtained to a precision of about±10%, and the work-hardening rate to about ±25%. Such inferred constitutive relations are more likely to be reliable if the com-parisons are made in regimes within which creep does not significantly influence the behaviour, and in general the timescale of mea-surement is important.� 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Nanoindentation; Finite element analysis; Creep; Non-destructive testing

1. Introduction

Depth-sensing nanoindentation is commonly employedfor material characterization. For instance, the Young’smodulus can be determined from the unloading portionof the measured load–displacement curve, while hardnesscan be calculated from measured peak loads and residualindentation areas. Procedures have also been proposedfor the extraction of strain-hardening exponents, fracturetoughness values, viscoelastic properties and creep param-eters, although most of these incorporate gross simplifica-tions and are in general of doubtful reliability. It is ofparticular interest to obtain constitutive relations, i.e.stress–strain curves, r ¼ f ðe; _e; T Þ. This represents a majorchallenge, since the stress and strain fields beneath anindent are relatively complex (even for simple-shaped ind-enters), and the raw data obtained (primarily load–dis-

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doi:10.1016/j.actamat.2010.02.031

* Corresponding author. Tel.: +44 1223 334332; fax: +44 1223 334567.E-mail address: [email protected] (T.W. Clyne).

placement plots) cannot readily be interpreted so as toreveal constitutive relations.

In view of the complexity of these stress and strain fields,it is clear that comprehensive analysis, almost certainlyinvolving finite element method (FEM) modelling, is likelyto be essential. FEM simulations [1,2] have demonstratedthat it is possible, using pre-specified constitutive relations,to obtain predicted load–displacement curves that agreequite well with those obtained experimentally. Unfortu-nately, what might be termed the inverse problem – i.e.the inference of constitutive relations from observed behav-iour during indentation – is much less tractable. The under-lying factors responsible for this are:

(i) the relatively low sensitivity of measurable behaviour(such as load–displacement plots) to the characteris-tics being sought;

(ii) the fact that these characteristics incorporate severaldegrees of freedom (e.g. unknown values of yieldstress and work-hardening rate as a function of strain

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and, in some cases, strain rate, plus the possibility ofcreep effects); and

(iii) the relatively high sensitivity of measurable behav-iour to extraneous factors, such as physical boundaryconditions (e.g. frictional effects [3], surface layers,material anisotropy [4,5], tip shape imperfections[6], etc.).

While not expressed in quite these terms, the recent sur-vey of Guelorget and Francois [7] highlighted this issue ofsensitivity as being mainly responsible for the low accuracyand reliability of extracted constitutive relations obtainedhitherto.

Nevertheless, there have been several recent studies [7–14]of the inverse problem, and in principle it is readily tackled,i.e. the input constitutive relations are adjusted until goodagreement is found with experiment (commonly load–dis-placement curves). One issue is whether this adjustment pro-cess can be optimized in some way. Bouzakis and co-workers[9,10] presented their “Fast Approach of stress–strain curvesbased on naNOindentationS” (FANOS) algorithm, which isdesigned to do this and is claimed to be efficient in terms ofcomputing time. Lee et al. [11] suggested using numericalregression analysis, focusing on yield stress and a strain-hardening exponent as the key property parameters.

However, Dao et al. [8], and subsequently Pelletier [12],recognized the problem of sensitivities as being the under-lying cause of poor reliability and accuracy. Pelletierfocused on the role of indenter shape. Of course, one wayof addressing the relatively low sensitivity of the experi-mental response to the constitutive relation is to broadenthe range of experimentation, and carrying out tests onthe same material with several different indenter shapes isclearly one possibility. This was in fact the approachadopted by Heinrich et al. [15] in their parametric FEMstudy. They concluded that using information obtainedfrom two indenters led to more accurate estimates of theYoung’s modulus, yield stress, and strain-hardening expo-nent. Pelletier also suggested that extending the compari-sons to include predicted and measured residual indentshapes would boost the sensitivity and aid convergence tothe optimal constitutive relation. In his study, two differentuniaxial stress–strain relationships – (i) an elastic–plasticsolid and (ii) an elastic–plastic linear strain-hardening solid– gave almost identical indentation load–displacementcurves, while the residual indent shapes differed markedly(pile-up for case (i) and sink-in for case (ii)). This possibil-ity was also noted earlier by Liu et al. [16].

Other workers [13,14] have focused on the most appropri-ate functional form for the constitutive equation. However,it is clear that such concerns do not address the underlyingsources of error, although it is certainly worth noting thatthe operative deformation mechanisms, and their relationto postulated analytical equations, are of significance. Forinstance, in some cases the indent is confined to a single grainwithin a specimen, leading to anisotropy associated with slipsystem orientations [4,5,16,17]. As an example, microinden-

tation experiments performed on body-centred cubic (bcc)single crystals of W and Mo ((1 0 0), (0 1 1), and (1 1 1) faces)indicate that pile-up is sensitive to slip system orientation[18]. The same is true for face-centred cubic (fcc) crystals,although Lim and Chaudhri [19] noted that the indentationhardness values obtained from two grains of different orien-tation ((1 1 0), (1 1 1)) in a polycrystalline copper samplewere very similar. Nevertheless, the effects of crystallo-graphic orientation on indentation response (i.e. load–dis-placement curves and residual indent topography) havenot yet been fully investigated, although the approachadopted by Liu et al. [17] appears promising. In their study,load–displacement and residual indent data were obtainedfor single crystal copper in three orientations ((1 0 0),(0 1 1), and (1 1 1) faces), using a spherical indenter. Thesemeasured data were compared to FEM predictions obtainedusing a large deformation crystal plasticity user subroutine,incorporating the geometry of crystallographic slip. Correla-tions were established between the anisotropic nature of thesurface topographies and the operative, orientation-depen-dent slip systems.

Of course, constitutive relations obtained in this wayrelate to the single crystal (and its particular orientation)only. In practice, it is much more likely that constitutiverelations will be required for polycrystals, and, indeed, fac-tors related to the polycrystalline nature of the material,such as the grain size and the texture, will affect the consti-tutive relation. It is therefore clear that, in order to charac-terize the behaviour of polycrystalline materials in thisway, regions must be indented which contain more thanjust one or two grains.

Creep and other time-dependent phenomena also com-monly occur during indentation [20–26]. It is sometimesassumed that creep effects are negligible (at room tempera-ture), but in practice they are often highly significant, atleast for metals. Indentation tends to produce relativelyhigh local stresses and it is fairly common experience forthese to generate obvious time-dependent effects, such asprogressive indenter penetration when a constant load ismaintained. It therefore seems likely that, at least in manycases, it will be necessary to take account of the possibilityof creep affecting the results when experimental nanoinden-tation data are to be used to obtain constitutive relations.Of course, it may also be possible to obtain creep parame-ters from nanoindentation data, although this presentsmajor challenges and there has certainly been very littlesuccess in this area so far [23].

Finally, it is becoming clear that friction between inden-ter and specimen can affect the observed behaviour. Theanalyses of Mesarovic and Fleck [27] suggest that pile-upformation is likely to be inhibited by friction. The stress fieldbeneath the indenter was also reported to be affected byfriction. Mata and Alcala [28] modelled the effect of theindenter–specimen friction coefficient on: (i) hardness, (ii)surface deformation (i.e. pile-up or sink-in) and (iii) load–displacement curve. They noted that friction was importantfor solids that would ordinarily exhibit pronounced pile-up

J. Dean et al. / Acta Materialia 58 (2010) 3613–3623 3615

(i.e. those exhibiting low strain-hardening exponents, or lowwork-hardening rates), since it acted to oppose slippage atthe interface. For solids that would ordinarily exhibit onlymoderate pile-up (or even sink-in), on the other hand, fric-tion was less significant, since slippage at the interface wouldbe limited in any event. In these cases, the load response andthe plastic strain field remained relatively unaffected. How-ever, the FEM simulations carried out by Antunes et al. [29],concerning Vickers indentation into AISI M2 steel (with alow strain-hardening exponent of 0.01), for three frictioncoefficient values, gave predicted load–displacement plotsthat were indistinguishable. This appears inconsistent withthe above rationale, although it is possible that all of thecoefficient values used were large enough to eliminate inter-facial slippage. This is clearly an area requiring furtherstudy, particularly since there has been virtually no experi-mental work in which friction coefficients have been variedor measured.

In summary, while material constitutive relations can, inprinciple, be obtained using FEM modelling and experimen-tal nanoindentation data, in reality it has proved difficult toemploy such procedures with any confidence, since experi-mental data have tended to be relatively noisy and similarlevels of agreement between prediction and experiment canoften be obtained using a wide range of constitutive rela-tions. Developing a more robust procedure requires twotypes of improvement: (i) broadening of the range of exper-imental data being used to obtain convergence to a uniquesolution and (ii) reducing the errors caused by inadequatecapture of the effects contributing to the observed behaviour.This paper is aimed at making progress on both fronts.

2. Experimental procedures

2.1. Material

The experimental work in this study has been carriedout using extruded OFHC copper bar in its as-received

Fig. 1. Optical micrographs of the as-received copper, at: (a) low and (b) high m10 lm radius) spherical indenter loaded so as to penetrate to an initial depth

form. Specimens for metallographic investigation werecut from the bar using electric discharge machining(EDM). Samples were mechanically polished using conven-tional procedures. The specimen surface was etched withdilute ferric chloride. A micrograph of a polished andetched surface is presented in Fig. 1a. It can be seen thata range of grain sizes is present. Some are larger than thediameter of the indenter tip used in the current work (nom-inally 20 lm), while some are smaller. In order to obtain apolycrystalline response, data from indents such as thatshown in Fig. 1b have been studied, since it straddles sev-eral grains. It was noted that, when this was the case,reproducible load–displacement plots were obtained duringindentation, whereas if smaller indents were produced(within single grains), then much greater variations wereobserved in the load–displacement plots (see Section 2.3below).

2.2. Macroscopic mechanical testing

2.2.1. Plastic deformation

Macroscopic mechanical properties were measuredusing conventional mechanical test procedures, focussingmainly on the extrusion direction, which was also the direc-tion of indentation. Cylindrical specimens (6 mm diameterand 8 mm height) were tested in compression, using a10 kN Instron hydraulic mechanical test machine, underdisplacement control, with a displacement rate of0.001 mm s�1 – i.e. an initial strain rate of 2 � 10�5 s�1.Compression tests were conducted along the extrusion axis.The specimens were located between flat platens of siliconcarbide. The ends were lubricated with molybdenum disul-phide, to minimize barrelling. Displacements were mea-sured using an LVDT.

2.2.2. Creep behaviour

Specimens for creep testing were CNC-machined fromthe copper rod, and loaded in uniaxial tension along the

agnifications, with the latter including an indent produced by a (nominallyof 1.65 lm.

Fig. 2. Sets of load–displacement plots, obtained using an initialpenetration depth of: (a) 0.4 lm and (b) 1.65 lm. In the first case, all ofthe indents were located within single grains, while in the second case theyall straddled several grains. The plot with the thicker (blue) line in (b)corresponds to the indent used for several comparisons in the currentpaper between measured and predicted behaviour. (For interpretation ofthe references to colour in this figure legend, the reader is referred to theweb version of this article.)

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extrusion axis. Testing was carried out using a series ofstresses and temperature (r = 50, 150 and 250 MPa, andT = 291, 373 and 473 K), under load control (with a weightacting under gravity). Displacements were measured usingclip gauges.

2.3. Indentation procedure

Indentation testing was carried out using a pendulum-based nanoindenter – the Nanotest 600, supplied byMicroMaterials Ltd. Tests were carried out using a spher-ical diamond indenter (nominally 20 lm diameter), withthe indentation direction parallel to the extrusion axis ofthe copper bar. The testing was carried out under load con-trol. All tests were carried out at ambient temperature(291 K ± 2 K). For most cases, the loading rate was fixedat 10 mN s�1. This progressive increase in the load wasarrested when a depth of 1.65 lm had been attained. Theperiod required to reach this depth, for a given material,was therefore dependent on the material response. Oncethe specified penetration depth had been reached, the loadat that point was held constant for a specified period – inthe present case, this was 50 s. The introduction of such a(dwell) period is common practice, designed to help ensurethat the unloading response is purely elastic (and can thusbe used to measure the elastic modulus). Part-way throughthe subsequent unloading (also at 10 mN s�1), a furtherhold period of 30 s was introduced, in order to check forany thermal drift. Similar operations were also carriedout using a slow loading rate (0.5 mN s�1). The systemcompliance was established using high load indentationson three reference materials, namely single crystal tungsten,a tool steel and fused silica. Data are presented after cor-rection for this compliance.

The effect of the indent being confined to a single grain,as opposed to straddling several grains, is illustrated inFig. 2. The load–displacement plots in Fig. 2a correspondto 10 indents made with a penetration depth of 0.4 lm, giv-ing an indent diameter of about 5 lm, each of which waslocated within a single grain. The 20 plots shown inFig. 2b, on the other hand, were made to a depth of1.65 lm, giving a diameter of about 17–18 lm, each ofwhich straddled several grains. It can be seen that the singlegrain plots exhibit considerable scatter, arising from theeffect of grain orientation, whereas those of multiple grainindents are much more reproducible. One of the plots inFig. 2b, i.e. the one having a thicker (blue) line, corre-sponds to an indent which forms the basis of comparisonsin the present paper. It should, however, be noted that,while the scatter is relatively small, and the chosen plot isa “representative” one, it would not be justifiable to makeany deductions dependent to a precision much better than,say, ±5% on the absolute values of load or displacement.This level of variability is probably inherent in the degreeto which the response of a small volume can reflect thatof the bulk, although there may be a dependence on thegrain structure and texture.

The thermal drift rate was systematically monitored forthe 20 indents represented in Fig. 2b. The average rate wasabout 0.045 nm s–1. This is clearly a relatively low ratesince, over the period of a typical test, it would aggregateup to just a few nm. This is certainly resolvable, but noneof the deductions being made in the current work actuallydepends on resolutions of this order. Of course, if the driftcan be assumed to be linear during the test period, thenappropriate compensation can be made.

2.4. AFM measurements on indenter tip and residual indents

The indenter tip was characterized using a VeecoDimension 3100 AFM system, with an XY closed-loopscanner using TappingModee with RTESPs (rotated tap-ping etched silicon probes). After locating the surface andapex of the indenter tip, 512 � 512 pixel scans of 15 lm

J. Dean et al. / Acta Materialia 58 (2010) 3613–3623 3617

by 15 lm were performed. Projected area functions wereextracted from the data, using height histograms. Theresidual indent shapes were also characterized using theVeeco Dimension 3100 AFM system. After locating anindent, a 40 lm by 40 lm scan was taken. Analysis of thedata was carried out using WsXM software from Nanotec[30].

3. FEM model formulation

3.1. Meshing of the specimen and mechanical boundary

conditions

An axi-symmetric model was employed in ABAQUS/CAE. In fact, even if the indenter itself is a perfect sphere,which is not usually the case, such indents do commonlyexhibit at least some asymmetry, even when indenting poly-crystals. However, the error introduced by this is thoughtto be small in the current work. The implicit solver wasused for all simulations. The specimen was modelled as adeformable body and meshed with 5625 linear quadrilat-eral (hybrid) elements (type CAX4H). The hybrid elementformulation is recommended [31] for incompressible mate-rials and for cases when deformation is dominated by plas-tic flow. The mesh, which is shown in Fig. 3, was refineddirectly beneath the indenter, in order to improve the reso-lution in this region. A sensitivity analysis confirmed thatthis mesh was sufficiently fine to achieve convergence,numerical stability and mesh-independent results.

Fig. 3. The (axi-symmetric) FEM mesh employed in the model: alsoshown are a 10 lm radius sphere (blue, upper curve) and an indentersection obtained from AFM measurements (red, lower curve). (Forinterpretation of the references to colour in this figure legend, the reader isreferred to the web version of this article.)

It is common practice to model the indenter as an ana-lytical rigid body of perfect sphericity. However, it shouldbe noted that it is possible for the shape of such tips todeviate significantly from that of a perfect sphere. Forinstance, local defects may be present as well as systematicshape anisotropy (which can arise during machining fromhardness differences in different crystallographic direc-tions). Local defects probably just introduce a small levelof noise into the measurements – this was the conclusionof Chen and Chang [6] – but systematic distortions of crys-tallographic origin may be worth noting. Of course, theyare difficult to predict, but it may in some cases be justifi-able to carry out topographic profiling of the tip and incor-porate this measured shape into the FEM model. In thesimulations presented here, however, the indenter wasmodelled as an analytical rigid body, with dimensions con-sistent with the AFM-determined area function (Section2.4). This is often a worthwhile exercise, and in the presentcase it turned out that the AFM-determined shape differedsignificantly from that of an ideal 10 lm radius sphere.This is apparent in Fig. 3, in which the AFM-determinedshape is compared to that of the ideal sphere.

It is also important to consider carefully how the experi-mental conditions are being controlled during indentation,since precision is needed, not only in the acquisition of exper-imental data, but also in the way that the model is formu-lated. In the present case, the indenter load history(measured experimentally) was specified as a model bound-ary condition. In fact, it is probably more common for themaximum indenter displacement to be specified. In that case,the indenter load history can be predicted, for a given consti-tutive relation, and compared with the experimental data. Inthe present case, however, since the (measured) load historyhas been specified as input data for the model, a comparisonbetween measured and predicted displacement–time curvesis more meaningful, since both are effectively outcomes ofan “experiment” (practical or modelling).

3.2. Constitutive relations for material plasticity

It was assumed in the present work that the materialexhibited linear work-hardening (see Section 4.1). This typeof behaviour can be represented by an equation of the form

r ¼ rY þ Kep ð1Þwhere r is the stress, rY is the yield stress, ep is the equiv-alent plastic strain and K is the work-hardening rate. In or-der to explore the sensitivity of predicted behaviour to thework-hardening rate K, and the yield stress rY, five linearconstitutive relations have been employed.

3.3. Simulation of creep deformation

ABAQUS provides a general framework for definingtime-dependent viscoplastic behaviour via the user subrou-tine CREEP. This is based on the standard steady statecreep rate equation

3618 J. Dean et al. / Acta Materialia 58 (2010) 3613–3623

dedt¼ Arn exp

�QRT

� �ð2Þ

in which A is a constant, n is the stress exponent and Q isthe activation energy. Increments of creep strain, in a giventime increment, were calculated using this expression, tak-ing no account of the prior strain history of the volume ele-ment concerned. Values of n, A and Q were estimated fromexperimental data obtained during the macroscopic creeptests (Section 2.2.2). The simulations were thus based onthe assumption that only steady state creep is exhibited,and that, if a change in stress level (or temperature) occursin a particular region, it immediately deforms at the (steadystate) rate corresponding to these new conditions, with nodependence on prior deformation history. As has beenhighlighted by Goodall and Clyne [23], such neglect of pri-mary creep, and indeed of any prior strain history effects,may introduce substantial errors during simulation ofindentation.

3.4. Simulation of friction between indenter and specimen

The effect of friction on the indentation response wasinvestigated by incorporating a friction coefficient (inde-pendent of slippage rate and temperature) into the FEMmodel. This introduces a shear force opposing interfacialsliding, given by the product of the normal force and thefriction coefficient. (Unless otherwise stated, the value ofthe friction coefficient was zero, i.e. friction was neglected.)

4. Experimental data, model predictions and sensitivity

analysis

4.1. Macroscopic plasticity

An experimental stress–strain curve, obtained duringcompressive loading in the axial (extrusion) direction, isshown in Fig. 4. It can be seen that the yield stress is about

Fig. 4. Experimental stress–strain data for axial compression, togetherwith the five different constitutive relations used in the FEM model.

280 MPa and, after an initial transient, the work-hardeningrate is linear, at �100 MPa. Also shown in this figure areplots corresponding to five different constitutive relations,all with linear work-hardening rates. Relations LWH1-3are based on a yield stress of 280 MPa, with work-harden-ing rates of 100, 50 and 0 MPa, respectively, while LWH4and LWH5 are both based on work-hardening rates of100 MPa, with yield stresses of 200 and 360 MPa, respec-tively. It can be seen that the LWH1 relation was designedto represent the observed experimental behaviour.

4.2. Creep deformation

Experimental (steady state) strain rate data are plottedin Fig. 5a, as a function of stress, together with curvesobtained using Eq. (2), for the three temperatures con-cerned. These curves correspond to the (best fit) values ofA, n and Q shown in the figure. It is difficult to comparethese values with expectations for copper, since creepbehaviour is often quite sensitive to microstructure, andin any event there is little information in the literature for

Fig. 5. Experimental creep data, showing: (a) steady state creep rates, as afunction of applied stress and temperature, together with predictions fromEq. (2), obtained using the values of the creep parameters shown and (b) acreep strain history, for an applied stress of 250 MPa at roomtemperature.

Fig. 6. Experimental data for load and displacement histories duringindentation with two different loading rates. In both cases, a dwell periodwas introduced at the peak load and at a lower load.

J. Dean et al. / Acta Materialia 58 (2010) 3613–3623 3619

the creep of copper over this temperature range. The valueof Q (35 kJ mol�1) is well below the activation energy for(lattice) diffusion in copper, but at these relatively low tem-peratures it is likely that fast diffusion paths, with loweractivation energies, would predominate in the rate-deter-mining processes.

In general, these values are plausible, and give fairlygood agreement with the experimental data. Of course,they relate only to steady state behaviour, and give littleor no information about the primary creep characteristics.As expected, there is a substantial variation in creep rateduring the period prior to establishment of a steady state.This is illustrated by the creep strain plot shown inFig. 5b, which refers to room temperature and an appliedstress of 250 MPa. It may be noted that this plot is partic-ularly relevant to the indentation experiments carried outin the present work, which were all done at room temper-ature and generated local stress levels of approximately thismagnitude (i.e. around the yield stress) in significant vol-umes throughout the tests. The substantial differencebetween creep rates in primary and secondary (steady state)regimes can be noted by comparing initial and final gradi-ents. For example, in a period of 100 s, a strain of about10 lstrain would occur in the primary regime, while inthe secondary regime only about 0.1 lstrain would result.This factor of 100 between these creep rates is obviouslylikely to result in large underestimates of the creep duringindentation, if steady state rates are employed. Of course,in reality any particular volume element is likely to be ina condition corresponding to some intermediate pointalong the curve and the situation is further complicatedby the fact that the stress in the element may be changingthroughout the process. Nevertheless, the possibility thatuse of steady state creep rate data may give rise to substan-tial underestimates of the creep strain should be borne inmind.

4.3. Indentation

4.3.1. Experimental data

During indentation, load–displacement–time data werecontinuously recorded. Typical data are presented inFig. 6, which shows load and displacement histories fortwo different loading rates. The load history was in effectpre-specified (although the peak load was dependent onhow the material deformed), while the response of thematerial is reflected in the displacement history. It is clearfrom this figure that time-dependent deformation (i.e.creep) is influencing the observed behaviour, at least forthe lower loading rate, since the peak load (needed to cre-ate the specified indentation depth of 1.65 lm) is about13% lower in that case. It follows that a significant propor-tion of the deformation that has occurred at the lowerloading rate has taken place via creep mechanisms. Thishighlights the fact that, even at room temperature, it maybe necessary to incorporate creep effects into this type of

modelling, at least for many materials (metals) and loadingconditions.

4.3.2. Predicted displacement histories

Predicted displacement–time plots, obtained using dif-ferent constitutive relations, are compared with experimen-tal data in Fig. 7, for high and low loading rates. There areseveral noteworthy features. Consider first the high loadingrate case (Fig. 7a). The gradient (dd/dt) during loading ispredicted to exhibit some sensitivity to the work-hardeningrate (compare LWH1-3), although the differences are smallconsidering that the value is being varied from 0 (LWH3)to 100 MPa (LWH1). This gradient is more sensitive tothe yield stress (compare LWH1, LWH4 and LWH5).However, the differences between the predictions aregreater in terms of the displacement depth at the end ofthe loading period, or, equivalently, the load needed toreach a specified depth. It is clear that LWH1, which isthe one designed to represent the macroscopically mea-sured behaviour, gives good agreement in terms of the load(i.e. the time) needed to generate the specified displacementof 1.65 lm. It would certainly have been picked in prefer-ence to LWH4 and LWH5, on the basis of this plot, i.e.the yield stress could have been established fairly accu-rately in this way. The outcome is less sensitive to thework-hardening rate, which is unsurprising in view of thefact that only a relatively small volume of specimen is expe-riencing large plastic strains. Nevertheless, the LWH1 pre-diction appears to be somewhat closer to the experimentalplot than LHW2 or LHW3, although the caveat in Section2.3 about there being something like a ±5% uncertainty inthe experimental data (load, or time, in this case) should benoted.

However, it can also be seen in Fig. 7a that the creepcontribution to the behaviour during the dwell is not beingwell-captured – for example, the progressive penetration

Fig. 7. Comparison between experimental displacement history data andpredictions from the FEM model, for loading rates of: (a) 10 mN s�1 and(b) 0.5 mN s�1 (see Fig. 6), obtained using the creep parameters in Fig. 5and the constitutive relations shown in the legend.

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during the hold and the relaxation during the subsequentunloading period are both under-predicted. In fact, theoverall contribution of creep to the behaviour is relativelysmall for this high loading rate case, particularly duringthe loading phase, and this facilitates accurate evaluationof the constitutive relation. However, this is not the casefor the low loading rate experiment (see Fig. 7b). Creepis now making a major contribution to the behaviour, evenduring the loading phase, as was evident from the lowerload needed to generate the specified displacement(Fig. 6). That this creep contribution to the behaviour isbeing under-predicted by use of the steady state expression(Eq. (2)), using the parameters shown in Fig. 5a, is nowvery clear, since the LWH1 plot gives substantially lowerpredicted penetration depths than actually occurred duringthese experiments. If this comparison had been used toinfer the constitutive relation, then relatively large errorswould probably have arisen, with the under-predictedcreep deformation probably leading to anomalously lowvalues being inferred for the yield stress and the work-hard-ening rate.

The reason for the creep deformation being under-pre-dicted is not entirely clear. However, the most likely explana-tion lies in the neglect of primary creep. The assumption thatthe creep rate within any volume element instantaneouslyconforms to the steady state value for the stress level con-cerned, while mathematically tractable in a model of thistype, is clearly unrealistic. In reality, primary creep behav-iour, with creep rates substantially greater than those forthe corresponding steady state, may strongly affect, and evendominate, the overall behaviour. This would be consistentwith the suggestion of Goodall and Clyne [23]. If so, deduc-tion of (steady state) creep rate parameters from indentationdata will, to say the least, require some very careful measure-ment and analysis. Even taking account of the effect of creepfor the purposes of inferring constitutive relations mayrequire information about primary creep behaviour, ratherthan just steady state parameters. Of course, the obviousapproach to this problem is to use indentation data obtainedin regimes in which creep (steady state or primary) is notstrongly influencing the observed behaviour, although care-ful modelling and measurement may be needed in order toidentify these regimes with confidence.

4.3.3. Residual indent shape

A comparison is shown in Fig. 8 between predicted andmeasured residual indent shapes, for the high loading ratecase. The experimental plot represents a radial average ofAFM data. As with the predicted displacement histories,certain features are particularly sensitive to the constitutiverelations. In this case, the pile-up around the periphery ofthe indent is predicted to be greater when the materialexhibits less work-hardening and/or has a lower yieldstress. The predicted shape of the interior of the indent isalso sensitive to the work-hardening rate, with a highervalue giving shallower indents. The same is true of the yieldstress. It can be seen that the sensitivity of the indent shapeto yield stress and work-hardening rate is slightly differentfrom that of the load–displacement response, and this maybe helpful in attempting to converge on the “correct” con-stitutive relation. Of course, there are other, complemen-tary approaches to making the procedures morediscriminatory, such as using a range of tip shapes.

It can be seen in Fig. 8 that, of the constitutive relationsemployed, the “correct” one (LWH1) gives the closestagreement with the experimentally measured indent shape.Taken in conjunction with the corresponding displacementhistory comparison shown in Fig. 7a, which was notstrongly affected by creep (at least during the loadingphase), it is clear that the experimental indentation data,viewed in the light of FEM model predictions, can in thiscase be used to obtain a good estimate of the (correct) con-stitutive relation, at least if the assumption is made that thematerial exhibits linear work-hardening. This is clearly veryencouraging, although it may be noted that, in addition tocreep, there are other possible effects that might need to betaken into account, notably friction between indenter andspecimen (see Section 4.3.4).

Fig. 8. Comparison between experimental residual indent shape data, fora loading rate of 10 mN s�1 (Fig. 6), and predictions from the FEMmodel, obtained using the creep parameters shown in Fig. 4 and theconstitutive relations shown in the legend.

Fig. 9. Comparison, for a loading rate of 10 mN s�1 (Fig. 6), betweenexperimental data and predictions from the FEM model, obtained usingthe creep parameters shown in Fig. 5, the LWH1 constitutive relation andthe coefficients of friction shown in the legend, for: (a) the displacementhistory and (b) the residual indent shape.

J. Dean et al. / Acta Materialia 58 (2010) 3613–3623 3621

4.3.4. Effect of friction

A comparison is shown in Fig. 9 between the measureddisplacement history and residual indent shape and the cor-responding predictions, based on use of the LWH1 consti-tutive relation in conjunction with three differentcoefficients of friction. The “correct” value of the latter isnot really known, although it probably lies in the rangecovered by these simulations. It can be seen that relativelysmall changes in the value of the coefficient, within thisrange, can have a significant effect on both displacementhistory and indent shape. Ideally, therefore, it should bemeasured for the case concerned, or at least a value shouldbe employed which was obtained experimentally undersimilar conditions. Unfortunately, there are virtually nosuch data available at present. However, the sensitivity ofthe predictions to the value of the friction coefficient isnot very high, and the predictions obtained using a valueof zero actually seem to be closer than those for finite val-ues, so the issue does not appear to be of major concern. Ofcourse, one should again note that the experimental dataacquired during the test (time, or load, in the case ofFig. 9a) carry an uncertainty of ±5% just in terms of repro-ducibility. In any event, it would be very helpful if someexperimental measurements of the friction coefficient couldbe made, although evidently this is quite a challengingobjective.

4.3.5. Stress and strain fields

Fig. 10a shows that, for the reference model case, pre-dicted plastic strains after indentation range up to about90%, with significant residual strains extending a micronor two below the indenter. Predicted contours of von Misesstress, at peak load and after the indenter has beenretracted, are shown in Fig. 10b and c. Under peak load,the stress field extends to a depth of about 20 lm, withpeak stresses of about 350 MPa, while residual stresses

are created in a region extending about 10 lm below theindenter, with peak stresses of about 200 MPa. This peakstress under maximum load is broadly consistent with theLWH1 constitutive relation, for a strain approaching100%. However, it is worth emphasizing again that creepis being under-predicted during this modelling. In practice,even for the high loading (and unloading) rate, creep relax-ation may reduce the stresses below the levels shown inFig. 10c, by the time that unloading is complete, and it iscertainly clear that they will be appreciably reduced bycreep quite soon afterwards.

5. Conclusions

The following conclusions can be drawn from this work.

(a) Quasi-static nanoindentation has been carried out onextruded copper. The data obtained have been com-pared with predictions from an FEM model, usinga range of constitutive relations (one of which corre-

Fig. 10. Predicted fields for the reference case (LWH1, with no friction,the creep parameters shown in Fig. 5 and a loading rate of 10 mN s�1

(Fig. 6)), showing: (a) residual equivalent plastic strain, (b) von Misesstress at peak indentation depth, and (c) von Mises stress immediatelyafter unloading.

3622 J. Dean et al. / Acta Materialia 58 (2010) 3613–3623

sponded to the macroscopic experimental stress–strain curve) and a fixed set of (steady state) creepparameters (obtained experimentally). The objectivewas to explore how such comparisons can be opti-mized, with a view to using them to obtain constitu-tive relations from nanoindentation data.

(b) Good agreement was obtained between experimentaldata and model predictions for the load–displace-ment–time relationships and for residual indentshapes, using the experimentally obtained constitu-tive relationship, providing a relatively high loadingrate was used, such that creep was not strongly affect-ing the observed behaviour. The sensitivity to theconstitutive relationship in such cases, particularlyfor the load required to reach a given indenter depth,at a given loading rate, was quite high. On this basis,assuming the correct constitutive relation to be basedon a fixed yield stress and a constant work-hardeningrate, it would have been possible to establish thesevalues with reasonable accuracy via these compari-sons, using the data acquisition and modelling proce-dures described here. A rough estimate of theprobable accuracy with which the yield stress andthe work-hardening rate could have been estimatedin this way would be ±10% and ±25%, respectively.Of course, if it had not been known that the materialexhibited linear work-hardening, then the procedurewould have been slightly more complex and lessaccurate.

(c) For some loading regimes, however, such as when arelatively slow loading rate is employed, the effectof creep deformation is likely to be pronounced, obvi-ously depending on the creep characteristics of thematerial and the temperature. Even if this creepbehaviour were to be relatively well-captured, thiswould undoubtedly introduce errors into the proce-dure for deducing the constitutive relation fromindentation data. In fact, use of the experimentallyobtained (steady state) creep equation in the presentwork led to a substantial under-prediction of the con-tribution of creep to the overall deformation. This isprobably because primary creep was dominating theobserved behaviour, with the associated strain ratesbeing much higher than in the corresponding steadystate. In fact, the difference, for the regime of temper-ature and stress relevant to the indentation testing,was estimated from the creep testing data to be a fac-tor of approximately 100. Of course, the stress level inany particular region tends to change continuouslyduring indentation, inhibiting the establishment oftrue steady state creep, and there are always regions(peripheral to the plastic strain field) that are enteringthe primary creep regime. In general, it is clear thatconsiderable attention must be devoted to the possi-bility that creep deformation is occurring duringindentation experiments being used to deduce consti-tutive relations.

(d) The predicted effect of friction on the displacement–time response and residual indent shape has beeninvestigated. The data suggest that these responsesare expected to show at least some sensitivity to thefriction coefficient, over the range in which it isexpected to lie, although the best agreement between

J. Dean et al. / Acta Materialia 58 (2010) 3613–3623 3623

experiment and predictions was actually obtainedusing a value of zero. However, this is a rather tenta-tive conclusion, limited by the accuracy of the exper-imental data. There is certainly an incentive tomeasure friction coefficients during indentation,which presents certain experimental challenges.

Acknowledgements

The authors would like to acknowledge financial sup-port from EPSRC (Platform Grant). There has also beenassistance from Mr. Chris Dunleavy, Dr. Sandra Korteand Mr. Robert Stearn, all of the Materials ScienceDepartment in Cambridge.

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